Evaluate: ∫tan x/(1 - sin x)dx

Question

Evaluate:

\(\int\frac{tan\,x}{(1-sin\,x)}dx\)

∫tan x/(1 - sin x)dx

Answer 1

Let I = \(\int\frac{tan\,x}{(1-sin\,x)}dx\) = \(\int\frac{sin\,x}{cos\,x(1-sin\,x)}dx\)

Put t = sin x 

dt = cosx dx

Now Putting 1-t = 0 

t = 1 

A(0) + B(0) + C(1 + 1) = 1

C = 1/2

Now Putting 1 + t = 0 

t = -1 

A(2)2 + B(0) + C(0) = -1

A = - 1/4

By equating the coefficient of t²,we get, A - B = 0

-1/4 - B = 0

B = - 1/4

From equation(1),we get,